Diffusion-weighted magnetic resonance imaging (dMRI or DW-MRI) is a non-invasive imaging technique that provides an indirect measurement of the diffusion profile over a finite spatial grid of points called voxels (three-dimensional pixels). Assessment of the diffusion profile is particularly useful for the study of the white matter microstructure. Indeed, the white matter is partly constituted of axons that can be thought as cylindrically shaped cells with impermeable membranes. As such, water trapped within axons is subject to restricted diffusion mainly along the axons' axis. The axon-specific diffusion profile thus carries valuable information about the axon structure itself (e.g., orientation, diameter). However, the diffusion-weighted signal is measured at the scale of the voxel, the size of which—typically 2×2×2 mm3—defines the spatial resolution of the diffusion-weighted images. Due to hardware limitations of the MRI scanner, a voxel cannot be made as small as the cells to be characterized, but instead contains thousands of axons with possibly different orientations and might also contain other types of white matter cells. A common approach to account for that is to mentally regroup axons into bundles with common orientation, hereafter referred to as fascicles, and to model the diffusion profile at the voxel level as a mixture of diffusion profiles in multiple fascicles. This type of modeling is hereafter referred to as multi-compartment model (MCM), where each compartment represents a fascicle. Other diffusion profiles might be added in the mixture to account for freely diffusing water (water not trapped within cells) or water trapped in glial cells. Therefore, assessment of the voxel-dependent diffusion profile as an MCM provides valuable information about the tissue architecture, which may be of scientific and/or clinical relevance. DW-MRI of the brain is also at the basis of tractography, which is a 3D-modeling technique used to virtually represent neural tracts.
DW-MRI provides a collection of diffusion-weighted (DW) images, each one of them being obtained by the application of a magnetic field spatial gradient, hereafter referred to as a diffusion-sensitizing gradient (DSG), wherein the intensity of each voxel is proportional to how far water molecules in this voxel moved along the DSG direction during a given diffusion time, which is another imaging parameter. Diffusion can be probed at different length scales by varying the intensity of the DSG, set through a third imaging parameter called the b-value. From a collection of DW images, it is then possible to infer an MCM in each voxel and subsequently assess the underlying white matter microstructure.
However, due to the low spatial resolution of dMRI, —e.g. 2×2×2 mm3, as mentioned above—several fascicles often coexist within a single voxel of the white matter, hindering both tractography and the determination of white matter microstructure.
Multi-compartment model selection allows overcoming this difficulty.
The DW signal for a given DSG can be analytically related to the parameters of the MCM that describe the diffusion of water in the different fascicles, provided that the number of non-parallel fascicles (“compartments”) in the voxel is known a priori. The parameters of the model represent e.g. the fascicle orientations and occupancies (i.e. the fraction of the voxel volume occupied by each fascicle). Optimal values for these parameters can be determined by fitting the model to a collection of measured DW signals corresponding to different DSGs. However, the number of compartments is not known in advance in practice; therefore, determining the fascicle configuration within each voxel becomes a model selection problem, wherein both the optimal number of compartments and the parameters defining said compartments have to be estimated.
So far, this model selection problem has been solved either by brute-force methods or using Bayesian frameworks.
In brute-force approaches, a set of nested candidate MCMs with increasing number of fascicles is fitted to the measured DW signals. The best MCM is then identified as the candidate model that “best” fits the signals, where the comparison usually relies on an F-test [1]. Since the more complex the model, the better the fit, the F-test often tends to favor MCMs that over-fit the signals because the same signals are used for estimation and to assess goodness of fit. To limit overfitting, the Bayesian information criterion has been introduced to penalize model complexity that increases with the number of fascicles [2]. Document EP 2 458 397 suggests using the Akaike Information Criterion (AIC) for performing MCM selection. More recently, generalization error has been proposed to choose the “optimal” MCM based on its ability to predict new signals [3], thus avoiding the overfitting issue. These approaches limit their search of the “optimal” MCM to a predefined candidate set. Therefore, they do not make optimal use of the available information.
Differently, Bayesian frameworks try to estimate the “best” MCM as the one that maximizes a posterior distribution on the models. They rely on a careful choice of a prior distribution for MCMs. For instance, [4] uses Markov random fields (MRF) while [5] resorts to Automatic Relevance Detection (ARD) in which non-informative priors are assigned to all the MCM parameters except the mixture weights that are assumed to be Beta-distributed. Such priors automatically prune an entire compartment if it is not supported by the signals. These methods simultaneously perform model estimation and selection. When translated to clinics, however, Bayesian methods have limitations. First of all, they are prohibitively computationally expensive. Moreover, clinical DW-MRI often includes a single DSG intensity (“b-value”) and a set of 30 DSG directions [6]. With such small sample sizes, the posterior distribution strongly depends on the prior, making the Bayesian information updating potentially ineffective.